Weak approximation for stochastic differential equations with small noises

New approach to construction of weak numerical methods, which are intended for Monte-Carlo technique, is proposed for a stochastic system with small noises. The theorem on estimate of method error in terms of product hiε j (h is a time increment, ε is a small parameter) is proved. Various efficient weak schemes are derived for a general system with small noises and for systems with small additive and small colored noises. The Talay-Tubaro expansion of the global error is considered for such systems. The efficient approach to reduction of the Monte-Carlo error is proposed. The derived methods are tested by calculation of Lyapunov exponents and by simulation of a bistable dynamical system for which multiplicative stochastic resonance is observed